The timing is after the candidate has been "nominated," or after the last day for filing candidacy. This is before ballots are printed. The law gives the court 10 days within which to render a decision, and the court decision is directly appealable to the state supreme court.
Contrast with AZ law (as it stands now) regarding presidential nominees. The SoS certifies eligibility, and that decision may not be challenged.
My other point is that the AZ legislature can avoid all the minutia that pertains to defining NBC, and simply provide that after the SoS certifies "qualified," the public may challenge on substantive grounds.
3. The board of supervisors and the recorder of each county or the clerk of each city or town who is responsible for preparing the ballots that contain the challenged candidate's name.
Perhaps you can point me to the section that supports:
The SoS certifies eligibility, and that decision may not be challenged.
Thanks.